On The Bayesian Analysis of Censored Mixture of Two Topp-Leone Distribution

This paper develops a Bayesian analysis in the context of non-informative priors for the shape parameter of the mixture of Topp-Leone using the censored data. A population of certain objects is assumed to be composed of two subgroups mixed together in an unknown proportion. The random observation taken from this population is supposed to be characterized by one of the two distinct unknown members of a Topp-Leone distribution. We model the heterogeneous population using two components mixture of the ToppLeone distribution. A comprehensive simulation scheme has been carried out to highlight the properties and behavior of the estimates in terms of sample size, corresponding risks and the mixing weights. A censored mixture data is simulated by probabilistic mixing for the computational purpose. The Bayes estimators of the said parameters have been derived under the assumption of non-informative priors using different loss functions. Posterior risks of the Bayes estimators are compared to explore the effect of prior information and loss functions. Bayes estimators assuming the uniform prior have been observed performing better.


Introduction
Topp-Leone distribution (T-L distribution) is a continuous unimodal distribution with bounded support. It is a two-parametric family continuous distribution proposed by Topp and Leone (1955). Such a distribution is useful for modeling lifetime phenomena, different aspect of this class of distributions studied by Nadarajah and Kotz (2003). The Topp-Leone distribution does not seem to be very familiar to the statisticians and has not been investigated in much detail under the Bayesian paradigm. The purpose of this study is to obtain the estimates for the parameter assuming different asymmetric loss functions.
This distribution has attracted recent attention and some key references are Ghitany et al. (2005), Van Dorp and Kotz (2004), Zhoiu et al. (2006), Kotz and Seier (2007), Nadarajah (2009), and Genç (2012). As well as having finite support, the T-L distribution has a "J-shaped" density function and a hazard function that is "bathtub-shaped". The latter characteristic is especially important in reliability applications in a wide range of fields, as is discussed recently by Reed (2011).
In recent years, the mixture models have received a considerable attention in the area of survival analysis and reliability. Mixtures of lifetime distributions occur when two different causes of failure are present, each with the same parametric form of lifetime distributions. Titterington et al. (1985), Lindsay (1995), Mclachlan and Peel (2000), Mcculloch and Searle (2001) and Demidenko (2004) are amongst the authors considering the analysis of the mixture models. The characterizations of mixtures studied by Nassar (1988), Ismail and El Khodary (2001) focused on the mixing fraction of the Exponential mixture. Further a mixture of two inverse Weibull distributions discussed by Sultan et al. (2007). Sindhu  The article is outlined as follows. In section 2, we define the mixture model of Topp-Leone. Sampling and likelihood is developed in section 3. In sections 4, the expressions for the posterior distributions have been presented. The section 5 contains the derivation of the said estimators and corresponding posterior risks. Credible intervals are derived in sections 6. A simulation study is performed in Section 7. The real life data application is explained in Section 8. Some concluding remarks are given in section 9.

The Population and the Model
A population is postulated to be composed of two subpopulations with specified parameters. The subpopulations are mixed in proportion w, (1-w), where 0<w<1. A finite mixture distribution function with the two component densities of specified parametric form (but with unknown parameters,  12 and  and with unknown mixing weights, w and (1-w) is: The corresponding finite mixture density function has its probability density function (p.d.f) as:

The Maximum Likelihood Function
The likelihood function for a two-component mixture with n items under study, the probability that 1 r will fail due to cause 1, 2 r will fail due to cause 2 and remaining 12 () n r r  will survive at time T when test is terminated is given as n r n r k n r n r k r r k r kk
Assuming independence, we have an improper joint prior that is proportional to a constant. The joint prior is incorporated with the likelihood (2) to yield a proper joint posterior distribution of 12 The marginal distribution of each parameter is obtained by integrating out nuisance parameters.
n r n r k x n r n r k jj

Posterior distribution under Jeffreys' prior
Jeffreys' prior is locally uniform and hence non-informative. An appealing property of Jeffreys' prior is that it is invariant with respect to one-to-one transformations. For the Topp-Leone model given in Section 2, the Jeffreys' , 0 , ( ) , 0 and ( ) 1, 0 1.  (2) to yield a proper joint posterior distribution of 12 ,  and w . The joint posterior distribution under Jeffreys' prior is: The marginal distribution of each parameter is obtained by integrating out nuisance parameters.

Bayes Estimators and Posterior Risks
In order to select a best decision in decision theory, a suitable loss function must be specified. The preference of loss function is a difficult job, and its selection is often formed for the reasons of mathematical convenience without any particular decision problem of ongoing interest excluding cost effect. As in the risk analysis, the potentiality of undesired event and its consequences both are explored. This potentiality is usually measured by failure rate. The Bayesian approach is extensively applied to failure rate. In disastrous outcomes, it can be terrible to underestimate the potentiality of an event rather than to overestimate. This is significant when the risk level is the basis of risk reducing initiative, either by reducing the potentiality or the consequences. An inappropriately low estimate of the risk level can lead to the lack of necessary steps to reduce the risk level. Hence, it is unreasonable to use a loss function that allows the estimation of a failure probability of zero. A positive loss at the origin allows the estimation of zero and in risk analyses estimating a zero failure probability simply means that no risk is expected for further detail see  Norstrom (1996). Norstrom (1996) has introduced precautionary loss function and is defined as The credible interval for the Jeffreys' prior can be constructed similarly.

Simulation Study
A simulation study is carried out to investigate the performance of Bayes estimators and the impact of sample size and mixing proportion. We take random samples of sizes n = 50, 100, 200, 300 and 500 from the two component  Tables 1 to 6. Credible intervals are presented in Table 7 to 8. The comparison observed has been summarized in last section.

Real Data Example
This section includes the analysis of a real life data to illustrate the methodology discussed in the previous section. In order to discuss the practical applicability of the results obtained under above sections, the following real life data on Survival times for 30 light bulbs in months presented by Butler (2011) have been used for analysis. The idea has been to verify whether the outcomes and properties of the Bayes estimators explored by simulation study have the same behavior under a real life environment. It is assumed that mixing weight and censoring rate are 0.6 and 0.6, wT  respectively. Under these assumptions, the distribution of observed data into censored subsamples of size 12 16 and 12, rr  respectively. The Bayes estimates under type I mixture censored samples based on uniform and Jeffreys' prior using PLF, WSELF and APLF have been presented in the following tables.  prior. The performance of estimates under the assumption of Uniform prior is better than Jeffreys' prior under WSELF and APLF loss function. The Bayes estimates under Uniform prior are more precise than its non-informative counterparts. Also, the results, obtained through real life data reported in Table  9 coincide with the simulation results with few exceptions showing the correctness of the simulation scheme.
Tables 7-8, give the results of Interval estimation. The credible intervals work quite well under the Jeffreys' prior. The width of credible interval is inversely proportional to sample size. The findings of the present study suggest that in order to estimate 12 , and w , the use of asymmetric loss function under the uniformed prior can be preferred.